27 28 28 triangle

Acute isosceles triangle.

Sides: a = 27   b = 28   c = 28

Area: T = 331.1633007445
Perimeter: p = 83
Semiperimeter: s = 41.5

Angle ∠ A = α = 57.65108983208° = 57°39'3″ = 1.00661979924 rad
Angle ∠ B = β = 61.17545508396° = 61°10'28″ = 1.06876973306 rad
Angle ∠ C = γ = 61.17545508396° = 61°10'28″ = 1.06876973306 rad

Height: ha = 24.53105931441
Height: hb = 23.65545005318
Height: hc = 23.65545005318

Median: ma = 24.53105931441
Median: mb = 23.67548812035
Median: mc = 23.67548812035

Inradius: r = 7.98798315047
Circumradius: R = 15.98800457208

Vertex coordinates: A[28; 0] B[0; 0] C[13.01878571429; 23.65545005318]
Centroid: CG[13.67326190476; 7.88548335106]
Coordinates of the circumscribed circle: U[14; 7.70546649011]
Coordinates of the inscribed circle: I[13.5; 7.98798315047]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 122.3499101679° = 122°20'57″ = 1.00661979924 rad
∠ B' = β' = 118.825544916° = 118°49'32″ = 1.06876973306 rad
∠ C' = γ' = 118.825544916° = 118°49'32″ = 1.06876973306 rad

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How did we calculate this triangle?

Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 27 ; ; b = 28 ; ; c = 28 ; ;

1. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 27+28+28 = 83 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 83 }{ 2 } = 41.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 41.5 * (41.5-27)(41.5-28)(41.5-28) } ; ; T = sqrt{ 109668.94 } = 331.16 ; ;

4. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 331.16 }{ 27 } = 24.53 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 331.16 }{ 28 } = 23.65 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 331.16 }{ 28 } = 23.65 ; ;

5. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 28**2+28**2-27**2 }{ 2 * 28 * 28 } ) = 57° 39'3" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 27**2+28**2-28**2 }{ 2 * 27 * 28 } ) = 61° 10'28" ; ; gamma = 180° - alpha - beta = 180° - 57° 39'3" - 61° 10'28" = 61° 10'28" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 331.16 }{ 41.5 } = 7.98 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 27 }{ 2 * sin 57° 39'3" } = 15.98 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 28**2+2 * 28**2 - 27**2 } }{ 2 } = 24.531 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 28**2+2 * 27**2 - 28**2 } }{ 2 } = 23.675 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 28**2+2 * 27**2 - 28**2 } }{ 2 } = 23.675 ; ;
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